The table gives a particle's velocity $v(t)$ in meters per second. Use the table to approximate its acceleration at $t=3$ seconds.

The table gives a cart's position $s(t)$ in meters. Approximate the cart's velocity at $t=5$ seconds using the closest values on both sides of 5.

The table gives the velocity $v(t)$ of a drone in meters per second. Approximate its acceleration at $t=4$ seconds.

The table gives the amount $W(t)$ of water in a tank, in liters. Estimate the average rate of change of $W$ from $t=1$ to $t=4$ hours.

The table gives a runner's velocity $v(t)$ in meters per second. Which value best approximates the runner's acceleration at $t=7$ seconds?

The table gives the height $H(t)$ of a balloon, in meters. Estimate the balloon's instantaneous velocity at $t=3$ seconds.

The graph shows velocity $v(t)$ in meters per second. At which listed time is the particle moving in the negative direction and increasing its speed?

The graph shows velocity $v(t)$ in meters per second. At which listed time is the particle moving in the positive direction but slowing down?

The graph shows velocity $v(t)$ in meters per second. At which listed time is the particle moving in the negative direction and slowing down?

The graph shows velocity $v(t)$ in meters per second. At which listed time is the particle at rest?

The graph shows velocity $v(t)$ in meters per second. At which listed time is the particle moving in the positive direction with increasing speed?

The graph shows velocity $v(t)$ in meters per second. At which listed time does the particle change direction?

A reservoir's inflow rate is $R(t)$ liters per hour, where $t$ is hours after midnight. What does $R^{\prime }(6)=-15$ mean?

The mass of algae in a tank is $A(t)$ grams after $t$ days. If $A^{\prime }(12)=4.8$, what is the correct interpretation?

The temperature $T(t)$ of a chemical bath is in degrees Celsius and $t$ is in minutes. If $T^{\prime }(9)=-2.3$, what are the units of $T^{\prime }(9)$?

The position $s(t)$ of a cart is measured in meters, and $t$ is in seconds. What does $s^{\prime }(4)=-3$ tell you?

A bacteria culture has population $P(t)$ cells, where $t$ is hours. If $P^{\prime }(2)=1200$, which conclusion follows?

The volume $V(t)$ of gas in a chamber is measured in liters. If $V^{\prime }(15)=0$, what must be true at 15 minutes?

A spherical balloon's volume increases at $40$ cm³/s. At the instant its radius is $5$ cm, how fast is the radius increasing?

A circular oil spill has radius $r$ meters. Its area is increasing at $18\pi$ m²/min when $r=3$ m. How fast is the radius increasing?

Water fills a cylindrical tank of radius $2$ m at $8\pi$ m³/min. How fast is the water depth increasing?

A 10-meter ladder rests against a vertical wall. Its base moves away from the wall at $1.2$ m/s. When the base is $6$ m from the wall, at what rate is the height of the top changing?

A student writes: “Because $v(4)=0$, a particle changes direction at $t=4$.” The velocity is positive on both $(3,4)$ and $(4,5)$. Does the justification establish the claim?

A student claims that a differentiable function $f$ has a local maximum at $x=2$ because $f^{\prime }(2)=0$. Does this evidence prove the claim?

A student writes: “Since $R^{\prime }(3)>0$, the amount of water in the reservoir is increasing at $t=3$.” Here $R(t)$ is the rate water enters, in L/h. Is the reasoning valid?

A student claims a particle is speeding up at $t=6$ because $v(6)>0$. The acceleration satisfies $a(6)<0$. Does the justification establish the claim?

A tank's water depth is $h(t)$ centimeters. Given $h(10)=24$, $h^{\prime }(10)=1.6$ cm/min, and $h^{\prime \prime }(t)>0$, use the tangent line at $t=10$ to approximate $h(10.5)$ and classify the estimate.

The mass $M(t)$ of a sample is in grams. Given $M(4)=80$, $M^{\prime }(4)=-3$ g/h, and $M^{\prime \prime }(t)<0$, approximate $M(4.2)$ with the tangent line and classify the estimate.

The position of a probe is $s(t)$ meters. Given $s(2)=15$, $s^{\prime }(2)=4$ m/s, and $s^{\prime \prime }(t)>0$, use linearization at $t=2$ to estimate $s(1.8)$ and classify the estimate.

A culture's mass is $B(t)$ grams. Given $B(6)=52$, $B^{\prime }(6)=2.5$ g/day, and $B^{\prime \prime }(t)<0$, use the tangent line at $t=6$ to approximate $B(6.4)$ and classify the estimate.